Optimal. Leaf size=153 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt{b} x}\right )}{12 a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} a^{5/6} d}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{a}}\right )}{12 a^{5/6} d} \]
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Rubi [A] time = 0.0288537, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {394} \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt{b} x}\right )}{12 a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} a^{5/6} d}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{a}}\right )}{12 a^{5/6} d} \]
Antiderivative was successfully verified.
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Rule 394
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{-a-b x^2} \left (\frac{9 a d}{b}+d x^2\right )} \, dx &=-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{a}}\right )}{12 a^{5/6} d}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt{b} x}\right )}{12 a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} a^{5/6} d}\\ \end{align*}
Mathematica [C] time = 0.158877, size = 172, normalized size = 1.12 \[ \frac{27 a b x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right )}{d \sqrt [3]{-a-b x^2} \left (9 a+b x^2\right ) \left (27 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right )-2 b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{-b{x}^{2}-a}}} \left ( 9\,{\frac{ad}{b}}+d{x}^{2} \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} - a\right )}^{\frac{1}{3}}{\left (d x^{2} + \frac{9 \, a d}{b}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b \int \frac{1}{9 a \sqrt [3]{- a - b x^{2}} + b x^{2} \sqrt [3]{- a - b x^{2}}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} - a\right )}^{\frac{1}{3}}{\left (d x^{2} + \frac{9 \, a d}{b}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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